First principles investigation of halide based Rb2NaGaZ6 (Z = Br, I) double perovskites for energy harvesting applications

Extensive investigations have been conducted on the thermoelectric and optoelectronic characteristics of double perovskite compounds using the full potential linearized augmented plane wave (FP-LAPW) approach. Here we investigated Rb2NaGaZ6 (Z = Br, I) to explore its band structure, and electronic, optical and transport properties. Born's stability criteria have confirmed the mechanical stability of these compounds. Analysis of the elastic properties reveals their ductile nature, as indicated by a Poisson coefficient (υ) greater than 0.26 and a Pugh ratio exceeding 1.75 for Rb2NaGaZ6 (Z = Br, I). Computation of the bandgap values shows that both compositions possess a direct bandgap nature, with respective values of 2.90 eV and 1.25 eV. This suggests that substituting Br with I brings the band edges closer together, resulting in a decrease in the bandgap value. The optical properties are assessed based on the absorption coefficient, reflectivity, and dielectric constants. The thermoelectric properties, including thermal and electrical conductivities, power factor (PF), and figure of merit (ZT), are determined using the BoltzTrap code. The ZT values indicate that both compositions exhibit promising potential for various transportation applications.


Introduction
2][3][4] At the moment, to attain this energy demand, on a large scale, fossil fuels are being utilized which is consequently giving rise to environmental pollution and climate change.Therefore, to solve this issue renewable energy sources are required, 9,10 like solar panels that use the photovoltaic effect to transform solar energy into electrical energy. 11,12In the beginning organic-inorganic lead-based perovskites solar cells were being used on a large scale as they can reach a great efficiency of 25.2%. 5,6But soon they experienced two crucial issues: one is the chemical instability and the other is the toxic nature of lead which harms the environment and public health and retards its feasible use. 7,8The extensive interest of researchers in double perovskites has grown over the last few years due to their conceivable implementation in renewable energy. 13,14By using these perovskites the ongoing record power conversion efficiency reaches up to 25.7%. 157][18][19] Here A stands for alkali-alkaline earth metal and B and B ′ stand for transition or post-transition metals which lead to new technological innovations. 20The double perovskites were introduced at the beginning of the 1950s. 20In the recent past, many studies have been reported aer which, halide-based perovskites have achieved enormous attention, particularly for their applications in solar cells since they reveal direct band gaps, huge absorption coefficients in the UV range, and also big charge carrier mobility.The motive beyond the decision of choosing these particular compounds is their direct bandgap nature and utmost stable cubic structures.
In recent times, a lot of attention has been paid to exploring the potential of double perovskites for optoelectronic and thermoelectric devices.For instance, Rb 2 NaGaZ 6 (Z = Br, I) which was studied by Behera and Mukherjee in 2022 which is a cubic double perovskite and bandgap calculation reveals the bandgap value of 1.81 eV for Rb 2 InBiCl 6 and 1.32 eV for Rb 2 -InBiBr 6 with ductile nature.The compound Rb 2 InBiX 6 (X = Cl, Br) have a high value of power factor which exhibit that these compounds will be potential candidates for use in thermoelectric (TE) devices. 21Similarly, Mebed et al. in 2022 reported Rb 2 AgBiX 6 (X = Br, I) compound and noticed the bandgap value of 1.88 eV for Rb 2 AgBiX 6 and 1.22 eV for Rb 2 AgBiI 6 .These compositions also exhibited notable TE features for TE generators. 22In addition, Mahmood et al. in 2021 submitted their work on Rb 2 TeX 6 (X is replaced by Cl, Br, I) double perovskites that belong to space group Fm3m and possess a face-centered cubic structure.The bandgaps of these compounds were tuned from UV to visible region that is 3.2-1.80eV as X is replaced by Cl, Br, and I which makes them suitable for solarcell applications. 23In another report, Mahmood et al. in 2022 reported X 2 AgBiI 6 (X will be changed by K, Rb, and Cs) and noted the bandgap values as 1.35, 1.26, and 1.30 eV for X = K, Cs, and Rb, respectively.These bandgap values with suitable ZT make them ideal for solar cell applications and thermoelectric generators. 25In addition to halide-based double perovskites, oxides-based double perovskites also captivated considerable attention like Sr 2 BTaO 6 (B = Sb, Bi) was investigated by Manzoor et al. in 2022 and exhibited the indirect bandgap nature with bandgap values of 2.066 eV for Sr 2 SbTaO 6 and 0.972 eV for Sr 2 BiTaO 6 respectively.These materials are convenient for thermal devices and particularly Sr 2 BiTaO 6 is more favorable for IR devices. 24his comprehensive literature exposed that the structural exibility of double perovskites offers an extended bandgap tuning and allows them to x themselves for advanced technological applications.Since there is still plenty of room at the bottom available to be explored to check the feasibility of double perovskites for optoelectronic applications.Thus, in the present communication, we analyzed the optoelectronic and thermoelectric properties of Rb 2 NaGa(Br/I) 6 by utilizing the fullpotential linearized augmented plane wave approach using DFT.So, the optical properties that are determined by utilizing TB-mBJ disclosed the direct bandgap nature.Optical parameters like refractive index n(u), reectivity R(u), extinction coef-cient k(u), absorption coefficient a(u), optical conductivity s(u), and dielectric constants are also computed. 26By using BoltzTrap code the temperature and chemical potentialdependent thermoelectric properties are estimated.Our current investigation is considered to provide new contestants for conceivable implementation in the later-development of renewable energy devices.

Computational details
This study investigates the unique characteristics of double perovskite compounds, specically Rb 2 NaGaZ 6 (Z = Br, I) focusing on their structural, optoelectronic, and thermoelectric properties.These compounds exhibited a cubic structure with the Fm 3m space group.Our research employs the full-potential linearized augmented plane wave method (FP-LAPW) implemented through the WEIN2K code, which is based on Density Functional Theory (DFT) principles. 27,28The PBE-GGA approximation is utilized to determine the precise lattice constant of Rb 2 NaGaZ 6 (Z = Br, I).For investigating the structural properties and determining the exchange co-relation potential (V xc ), we employ the local density approximation (LDA) and the generalized gradient approximation (GGA).To accurately assess the bandgaps and band structures, we utilize the modied Becke Johnson (mBJ) approximation. 29,30In this method complete crystal is divided into two different categories related to the muffin tin, one is sphere and the other is an interstitial region, they are represented by the Fourier series.The input parameters for optimization and execution of SCF calculations are set as 1000 k-points of mesh size (12 × 12 × 12), angular momentum (l Max = 10), Gaussian factor (G Max = 16) and (R MT × K Max ) plane wave cutoff parameter = 8.
The thermoelectric properties are executed by the BoltzTrap code which is established on basis of Boltzmann Transport theory. 31Computation of thermoelectric characteristics including Seebeck coefficient (S), electrical (s), and also thermal (k) conductivities by utilizing the following mathematical forms: 32 Àvf m ðT; 3Þ v3
The PBEsol-GGA functional is employed for the analysis of structural parameters.By comparing the ionic radii of iodine (I) and bromine (Br) [Table 2], it is observed that the larger ionic radius of iodine leads to an increase in the lattice constant (a o ) from 10.79 Å to 11.81 Å.The calculated values of lattice parameters of Rb 2 NaGaBr 6 are in good agreement with the values reported at materials project database. 36The bulk modulus (B o ) indicates the material's ability to resist volume changes under applied pressure.The lattice constant, as well as Bulk modulus, are linked inversely thus as a o is increasing so the B o is assumed to decrease.Table 1 show a decrease in values of B o from 24.3 GPa and 19.46 GPa as I is replaced with Br.
To determine the stability of the crystal, Goldsmith tolerance factor t G is computed.For a stable structure, its value must be between 0.81 and 1.11.Our calculated values of the Goldsmith tolerance factors for double perovskites Rb 2 NaGaI 6 and Rb 2 -NaGaBr 6 are 0.94 and 0.95, respectively.A reduction in values of enthalpy of formation DH f was observed from −1.53 to −1.06 eV as composition is shied from Br to I. 37 The negative sign express the discharge of energy during the construction of these compounds which conrms the stability of the compound.To inspect the mechanical behavior of compounds the elastic constants C 11 , C 22 , and C 44 are computed. 38,39The Born Criterion condition shows as 40 They assist to nd out the estimation of the stiffness of compounds against strains.To examine if the nature of compounds is ductile or brittle the value of the B/G Pugh ratio is measured.The Pugh value of Rb 2 NaGaBr 6 (2.43) is higher than 1.75 and the value of the Poisson ratio (y) i.e., 0.32 is greater than 0.26 which reveals the ductile nature of this composition.However, when Br is replaced with I, the value of B/G and y are reduced to 1.50 and 0.22 which uncovers the transformation of composition from ductile to brittle nature.

Electronic properties
To check the nature of the compound whether it is metallic, insulator or semiconductor we calculated the carrier concentration and their bandgap values.In Fig. 3     database where GGA functionals is employed. 36As I has higher ionic radii than Br, which lowered the conduction band edge and thus reduced the bandgap value for Rb 2 NaGaI 6 .
In addition to this, the calculation of TDOS and PDOS was performed.The graphs depicting the density of states showcase the involvement of various states in the creation of energy bands, as indicated in Fig. 4. The TDOS graph reveals that the energy states in the valence band extend from −5.5 eV to the Fermi level.Furthermore, the conduction band edge is situated at 3.10 eV for Rb 2 NaGaBr 6 and 1.45 eV for Rb 2 NaGaI 6 .Examining the PDOS graph closely, it becomes apparent that the valence and conduction band formation heavily relies on the 4p states of Rb and Ga, 3s states of Na, and 4p and 5p states of Br and I.The p-states of halogen atoms predominantly contribute to the valence band maxima, while the 3s-states of Na contribute to the conduction band minima.Interestingly, replacing Br with I causes the states in the conduction band to shi closer to the Fermi level, resulting in a reduced band gap value.

Optical properties
In order to assess the signicance of Rb 2 NaGaZ 6 double perovskites (where Z represents Br or I) in the context of solar cell usage, we conducted a thorough analysis of their optical characteristics.The effectiveness of capturing and utilizing incident light in any optoelectronic apparatus relies heavily on interband electronic transitions.A part of incident light induces polarization in material whereas the rest of the light is scattered.To quantify these features the most important parameter is 3(u) dielectric constant which is utilized to optimize various parameters.
Here, 3 1 (u) shows the dielectric constant's real part and represents the polarization capacity of material in contrast to the dispersion of impinging light which is determined by the imaginary part of the dielectric constant 3 2 (u).The 3 1 (u) and 3 2 (u) are plotted for Rb 2 NaGaZ 6 (Z = Br, I) and presented in Fig. 5(a) and (b).On zero energy, 3 1 (0) shows the value of 2.58 and 3.4 for Rb 2 NaGaBr 6 and Rb 2 NaGaI 6 , respectively and that is called a static dielectric constant [see Table 2].Static dielectric constant and bandgap are varies inversely.The real part of the dielectric constant is obtained by employing Kramers-Kroning relation: P is representing the principal integral.
The understanding of 3 1 (u) and 3 2 (u) let us to calculate refractive index n(u), reectivity R(u), and absorption coefficient a(u).These relations can be investigated by utilizing the following equations: nðuÞ Incident energy lower than the bandgap is never able to generate electronic transitions from the valence band (VB) to the conduction band (CB).The value of total reection varied from Br to I as 0.05 to 0.08 respectively on 0 energy.The refractive index as well as a real part of the dielectric constant are connected by: Expand in the values of refractive index n(u) is witnessed as 1.61 to 1.84 for Rb 2 NaGaBr 6 and Rb 2 NaGaI 6 , respectively [see Table 2].
The coefficient of absorption evaluated the light energy that is absorbed by the semiconductor.The coefficient of extinction k(u) informs us of the attenuation of light.The light energy attenuation is indicated by a(u).The light energy that is captivated by the semiconductor is determined by the absorption coefficient a(u).Additionally, R(u) evaluated the reection and scattering of light also Fig. 5.At zero energy.At zero energy R(0) for Rb 2 NaGaBr 6 and Rb 2 NaGaI 6 is 0.05 and 0.08 respectively represented in Table 2.The variation of absorption coefficient a(u) with energy of the incident photons for Rb 2 NaGaI 6 and Rb 2 NaGaBr 6 are shown in Fig. 5(e).From this plot one can see that when the incident photon's energy is less than that of the band gap of the two materials, electromagnetic radiations are not absorbed.Based on the band gaps of these systems, Fig. 5(e) clearly shows the threshold in absorption for Rb 2 NaGaI 6 is at lower energies compared to Rb 2 NaGaBr 6 .Since only for the case of optical band gap a value of a(u) > 1.0 × 10 4 cm −1 is realized, 41 Fig. 5(e) clearly shows photons absorbed by the studied materials are in accordance with the band gaps of Rb 2 NaGaI 6 and Rb 2 NaGaBr 6 , indicating that these two materials are transparent to infrared photons.Compared to the case of hybrid perovskite materials, it can be seen that the optical absorption in the visible region for Rb 2 NaGaI 6 is week. 42However, higher absorption of visible light is seen for Rb 2 NaGaI 6 that can be assigned to the isolated conduction band at lower energies found for this material that causes absorption to begin at lower energies.

Thermal properties
The utilization of certain materials capable of converting waste heat into electrical energy is highly suitable for thermoelectric (TE) devices, making them excellent candidates for energy harvesting.To determine the thermal properties of these double perovskites, calculations were performed using the BoltzTrap code, which is based on transport theory and implemented through WEIN2k.The resulting gures, shown in Fig. 6(a)-(e), illustrate the electrical and thermal conductivities (s/s) and (k/ s), Seebeck coefficient (S), Power Factor (PF), and the gure of merit (ZT) across a temperature range of 200-600 K.A higher Seebeck coefficient indicates the favorable characteristics of a thermoelectric material.In order to achieve a greater ZT value, it is necessary for S, s, and PF to be higher.
A good TE material essentially has bigger values of parameters besides thermal conductivity.The electrical conductivity of each composition approximates the ow of charge.To recognize the sort of material in case it is a conductor, insulator, or semiconductor, the movement of charges is studied.For an outstanding TE device, the compound should essentially have a high estimation of (s/s).The estimation of the electrical conductivity reveals the potential of under-study compositions for designing new TE devices.Thermal conductivity map the ow of heat caused by the temperature gradient across the different ends of the material.This ow of heat could be either caused by electronic transportation called the electronic part of thermal conductivity (k e ) or lattice vibration called the lattice part of thermal conductivity (k l ).Rb 2 NaGaBr 6 has a higher slope than Rb 2 NaGaI 6 may be because electrons have the most energy.The determined smaller k e /s and greater s/s is in the privilege of TE efficiency.Fermi level takes place at the E g of the semiconductor also material application can be explained by free charge carrier which bounces from VB to CB.The Seebeck coefficient (S) which illustrates the fraction of potential difference induced due to temperature difference and suggested as thermo-power is indicated in Fig. 6.
It can be either positive or negative depending on the nature of p-type and n-type. 43As the temperature is increased the S reduces.The gure of merit can be narrated by: Generally, the TE performance of any composition is concluded by PF and ZT.The power factor measures the performance of the compound by a collaboration of s and S. The escalation in PF at higher temperatures is examined as it may cause the existence of a greater number of charge carriers because of the large atomic number.
The semiconducting materials with an excess of electrons and holes are termed as n-type and p-type semiconductors, respectively.Although in chemical potential p-type is represented the negative value and n-type materials can be witnessed by a positive value.When we change the temperature, a change    2.3 × 10 11 W K −2 m −1 s −1 along at 0 eV but as the temperature is increased up to 600 K a maximum curve is obtained at 6 × 10 11 W K −2 m −1 s −1 and in p-type doping, the highest peak is obtained at 4.1 × 10 11 W K −2 on 3.2 eV for Rb 2 NaGaBr 6 .For Rb 2 NaGaI 6 in n-type doping, the maximum is noticed at 2.1 × 10 11 W K −2 , at 0 eV the power factor has a maximum curve at 6 × 10 11 W K −2 and for p-type doping, the PF at 3.8 × 10 11 W K −2 is obtained along 1.8 eV.We witnessed by stats that as we increase temperature we can obtain the maximum value of power factor.p-type doping has maximum contributions.Investigated value of the gure of merit investigated at 0.78 eV shows an increase but as we increase the temperature the curve decreases.It was found to be unitary at 300 K at 0.01 eV.The thermal conductivities for n-type and p-type doping is noticed as 0.65 × 10 15 W m −1 K −1 s −1 and 0.80 × 10 15 W m −1 K −1 s −1 at 300 K for Rb 2 NaGaZ 6 (Z = Br, I) respectively.Fig. 8 shows the carrier concentration plot.As the Seebeck coefficient and carrier concentration are related inversely, as we increase the concentration S decreases.At 250 (mV K −1 ) the concentration is at −1.7 (e per uc) and as we can see by the graph that by further increasing the value of carrier concentration the value of Seebeck is dropping.In p-type doping we witness a maximum peak for Rb 2 NaGaI 6 at 850 (mV K −1 ) for 0.1 (e per uc) then it is decreasing linearly.The Power factor will give maximum value at a xed point aer that it will start declining.As PF depends upon S and s.For Rb 2 NaGaI 6 get maximum peak at 6.1 (mV K −1 ) for 0.6 (e per uc).The graph for the gure of merit depicts that as we increase the carrier concentration the ZT decreases.For Rb 2 NaGaBr 6 we received a unitary value at 0.01 ZT e and so for Rb 2 NaGaI 6 .Electrical and thermal conductivity has a direct relation with carrier concentration.As we start increasing value of Rb 2 NaGaBr 6 for s/s from 0 N (e per uc) at 1.8 × 10 −19 U −1 m −1 s −1 it starts increasing linearly.For Rb 2 NaGaI 6 the curve increases from 0.2 × 10 −19 U −1 m −1 s −1 .For k e /s as temperature goes on increasing from 300 K to 1200 K the concentration and thermal conductivity also increase.At 1.2510 15 W m −1 K −1 s −1 along 2 N (e per uc) we get the maximum curve.

Conclusion
In nutshell, we investigated the Rb 2 NaGaZ 6 (Z = Br, I) double perovskites to explore their potential for solar cell application.Opto-electronic and thermoelectric properties were computed by employing rst-principles computation using DFT-based WEIN2k soware.The structural stability of these compositions was calculated by measuring the value of enthalpy of formation which revealed that Rb 2 NaGaBr 6 was more stable with DH f = −1.53.Phase stability was assured by computed values of tolerance factor that was 0.95 and 0.94 for Br and Ibased compositions.The analysis of band structure exhibited the direct bandgap nature of both of these compositions and a signicant reduction in bandgap value from 2.90 to 1.25 eV upon replacement of Br with I.A low value of reectivity enlarges the potential of our studied compounds.Electrical and thermal conductivities were greater for Rb 2 NaGaBr 6 .While Rb 2 NaGaI 6 has a greater Seebeck coefficient.Better thermal efficiency with high absorption coefficient for UV photons make these compositions the best applicant for energy harvesting devices like solar cells.
Fig. 1 (a) Left side presents a cubic unit-cell ball and (b) the right side image shows polyhedral Rb 2 NaGaZ 6 (Z = Br, I) double-perovskites.
depicted the direct bandgap semiconducting nature of compound Rb 2 NaGaZ 6 (Z = Br, I) because the maxima of the valence band (VBM) and minima of the conduction band (CBM) lie at the same G point.The value of bandgap for Rb 2 NaGaBr 6 is 2.90 eV and 1.25 eV for Rb 2 NaGaI 6 noted from these band structures.It is clear that the band gap computed for Rb 2 -NaGaBr 6 using the TB-mBJ functionals gives larger value compared to the GGA values reported in materials project

Fig. 4
Fig. 4 Show the total and partial density of states for Rb 2 NaGaZ 6 (Z = Br, I).

Fig. 5
Fig. 5 Characterization of (a) real and (b) imaginary values of the complex dielectric function, In (c) the refraction and (d) the extinction coefficient (e) absorption and (f) reflectivity for Rb 2 NaGaZ 6 (Z = Br, I).

Fig. 7
Fig. 7 Rendering of Seebeck (S) coefficients, the power factor, the (ZT) figure of merit, electrical and thermal conductivity graphs contrary to the chemical potential for DPs Rb 2 NaGaBr 6 and Rb 2 NaGaI 6 .

Table 1
Values of lattice constant a o (Å), the bulk modulus B o (GPa), enthalpy of formation DH f (eV), the tolerance factor (t G ) for cubic Rb 2 NaGaZ 6 (Z = Br, I) double-perovskites

Table 2
Computed values of optical parameters at zero energy for Rb 2 NaGaZ 6 (Z = Br, I)